Moduli of representations of one-point extensions (Q6600730)

The authors study moduli spaces of (semi-)stable representations of one-point extensions of quivers by rigid representations. Let \(A=kQ\) be a path algebra and \(T\) be an \(A\)-module, denote by \(A[T]\) the one-point extension algebra.\N\NIn section 3, they first construct standard projective resolutions for representations of \(A[T]\). By using the resolutions, under the assumption of \(T\) being rigid, they prove the functor Ext\(^2\) vanishes on the so-called full modules (See Definition 3.5 and Theorem 3.6). They also calculate the Euler form of \(A[T]\) (See Corollary 3.10).\N\NFrom section 4, the authors consider representation varieties of \(A[T]\). They determine the Zariski tangent space of the representation variety in each point and conclude that the open subset of full representations is smooth and irreducible. Following the GIT approach of King in the construction of moduli spaces [\textit{A. D. King}, Q. J. Math., Oxf. II. Ser. 45, No. 180, 515--530 (1994; Zbl 0837.16005)], they find numerical criteria for non-emptiness and results on basic geometric properties, construct generating semi-invariants, expand the Gel'fand MacPherson correspondence, and derive a formula for the Poincaré polynomial in singular cohomology of these moduli spaces.